Generated Topologies
Definition
Suppose $\mathcal{T} _1$ and $\mathcal{T} _2$ are
two topologies on a set $X$.
Then $\mathcal{T} _1 \cap \mathcal{T} _2$ is a
topology on $X$.
More generally, the intersection of a family of
topologies is a topology.
Now suppose we are given some set of subsets $\mathcal{B} $ of $X$. The intersection of the set of topologies on $X$ which contain $\mathcal{B} $ is a topology on $X$. We call this the topology generated by $\mathcal{B} $.